Using the Matrix Sign Function to Compute Invariant Subspaces

The matrix sign function has several applications in system theory and matrix computations. However, the numerical behavior of the matrix sign function, and its associated divide-and-conquer algorithm for computing invariant subspaces, are still not completely understood. In this paper, we present a new perturbation theory for the matrix sign function, the conditioning of its computation, the numerical stability of the divide-and-conquer algorithm, and iterative refinement schemes. Numerical examples are also presented. An extension of the matrix-sign-function-based algorithm to compute left and right deflating subspaces for a regular pair of matrices is also described.